aequationes mathematicae

, Volume 54, Issue 1–2, pp 108–116 | Cite as

Fractal penrose tiles II: Tiles with fractal boundary as duals of penrose triangles

Research Papers

Summary

Suppose given a quasi-periodic tiling of some Euclidean space E u which is self-similar under the linear expansiong: Eμ→Eμ. It is known that there is an embedding of Eμ into some higher-dimensional space ℝ N and a linear automorphism\(\bar g:\mathbb{R}^N \to \mathbb{R}^N \) with integer coefficients such that E u ⊂ ℝ N is invariant under\(\bar g\) andg is the restriction of\(\bar g\) to E u .

Let E s be the\(\bar g\)-invariant complement of E u , and\(g* = \bar g^{ - 1} \left| {_{E_u } } \right.\). If certain conditions are fulfilled (e.g.\(\bar g\) must be a lattice automorphism,g * is an expansion), we construct a self-similar tiling of E s whose expansion isg *, using the information contained in the original tiling of Eμ. The term “Galois duality” of tilings is motivated by the fact that the eigenvalues ofg * are Galois conjugates of those ofg. Our method can be applied to find the Galois duals which are given by Thurston, obtained in a somewhat other way for the case that dim Eμ=1, andg is the multiplication by a cubic Pisot unit.

Bandt and Gummelt have found fractally shaped tilings which can be considered as strictly self-similar modifications of the kites-and-darts, and the rhombi tilings of Penrose. As one of the examples, we show that these fractal versions can be constructed by dualizing tilings by Penrose triangles.

AMS (1991) subject classification

05B45 52C20 52C22 

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Copyright information

© Birkhäuser Verlag 1997

Authors and Affiliations

  1. 1.Institut für Mathematik und InformatikUniversität GreifswaldGreifswaldGermany

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